Optimum linear coding for additive noise systems using information feedback

The feedback coding problem for Gaussian systems in which the noise is neither white nor statistically independent between channels is formulated in terms of arbitrary linear codes at the transmitter and at the receiver. This new formulation is used to determine a number of feedback communication systems. In particular, the optimum linear code that satisfies an average power constraint on the transmitted signals is derived for a system with noiseless feedback and forward noise of arbitrary covariance. The noisy feedback problem is considered and signal sets for the forward and feedback channels are obtained with an average power constraint on each. The general formulation and results are valid for non-Gaussian systems in which the second order statistics are known, the results being applicable to the determination of error bounds via the Chebychev inequality.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

Optimization and Control of Power Flow in Distribution Networks

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators.

Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory.

The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.

Incremental Control Synthesis for Robotics in the Presence of Temporal Logic Specifications

This thesis presents methods for incrementally constructing controllers in the presence of uncertainty and nonlinear dynamics. The basic setting is motion planning subject to temporal logic specifications. Broadly, two categories of problems are treated. The first is reactive formal synthesis when so-called discrete abstractions are available. The fragment of linear-time temporal logic (LTL) known as GR(1) is used to express assumptions about an adversarial environment and requirements of the controller. Two problems of changes to a specification are posed that concern the two major aspects of GR(1): safety and liveness. Algorithms providing incremental updates to strategies are presented as solutions. In support of these, an annotation of strategies is developed that facilitates repeated modifications. A variety of properties are proven about it, including necessity of existence and sufficiency for a strategy to be winning. The second category of problems considered is non-reactive (open-loop) synthesis in the absence of a discrete abstraction. Instead, the presented stochastic optimization methods directly construct a control input sequence that achieves low cost and satisfies a LTL formula. Several relaxations are considered as heuristics to address the rarity of sampling trajectories that satisfy an LTL formula and demonstrated to improve convergence rates for Dubins car and single-integrators subject to a recurrence task.